DSpace at VNU: On a backward Cauchy problem associated with continuous spectrum operator

Nonlinear Analysis 73 (2010) 1966–1972

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Nonlinear Analysis
journal homepage: www.elsevier.com/locate/na

On a backward Cauchy problem associated with continuous
spectrum operator
Nguyen Huy Tuan a,∗ , Dang Duc Trong b , Pham Hoang Quan a
a

Department of Mathematics, Saigon University, 273 An Duong Vuong, Ho Chi Minh City, Viet Nam

b

Department of Mathematics, University of Natural Sciences, Vietnam National University, 227 Nguyen Van Cu, Q.5, Ho Chi Minh City, Viet Nam

article

abstract

info

Article history:
Received 12 March 2009
Accepted 12 May 2010

The nonlinear backward Cauchy problem
ut + Au(t ) = f (u(t )), u(T ) = ϕ,
where A is a positive self-adjoint unbounded operator, which has a continuous spectrum

and f is a Lipschitz function being given is regularized by the well-posed problem. The new
error estimates of the regularized solution are obtained. This work extends to the nonlinear
case earlier results by the authors [7,1] and by Denche and Bessila [8,13].
© 2010 Elsevier Ltd. All rights reserved.

MSC:
35K05
35K99
47J06
47H10
Keywords:
Nonlinear parabolic problem
Backward problem
Semigroup of operator
Ill-posed problem
Contraction principle

1. Introduction
Recently, in 2008, in [1] we have considered the nonlinear parabolic backward in time problem
ut + Au = f (u(t )),

0 < t < T,

u(T ) = ϕ,

(1)
(2)

where A is a self-adjoint operator, having a discrete spectrum on a Hilbert space H, with the inner product , and the norm
, such that −A generates a compact contraction semi-group on H. This problem is well known to be severely ill-posed

and regularization methods for it are required. We have established, under the hypothesis that f is a global Lipschitzian
function from H to H, the existence of a unique solution for the approximated problem
ut + f (A)u = f (u (t )),

0 < t < T,

u (T ) = ϕ,

(3)
(4)

with 0 < < 1 and f (A) being chosen later under some better conditions. We assume that (1)–(2) has a unique smooth
solution u(t ). Under this assumption, we obtain an error estimate on the smoothing of u, generalizing a previous work of
in [2]. This error is given a form in Holder type
u(t ) − u (t ) ≤ M β( )t /T .

∗

Corresponding author.
E-mail address: (N.H. Tuan).

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2010.05.025


N.H. Tuan et al. / Nonlinear Analysis 73 (2010) 1966–1972

1967

We note that stability estimates of the Holder type for the nonlinear heat-parabolic equation backwards in time have been

established in some recent papers, such as [3–7,1]. These results give no information on the continuous dependence of the
solution on the data at t = 0. And so, the convergence of the approximate solution is very slow when t is in the neighborhood
of zero. To our knowledge, the case where the operator A has a discrete spectrum has been treated in many recent papers,
such as [8,9]. Once more, we also note that there are many works on the linear homogeneous case of the backward Cauchy
problem, but the literature on the nonlinear case of the problem is quite scarce. So, it is not easy to regularize the nonlinear
problems. Recently, the nonhomogeneous and nonlinear backward problem in Banach space has been considered by Hetrict
and Hughes [10,11].
As we know, the position operator usually has a continuous spectrum, much like the momentum operator in an infinite
space. But the momentum in a compact space, the angular momentum, and the Hamiltonian of various physical systems,
especially bound states, tend to have a discrete (quantized) spectrum—that is where the name quantum mechanics comes
from. The formal scattering theory has a strong overlap with the theory of continuous spectra.
In this paper, we shall use new methods to extend the continuous dependence results of [12,13] to more general nonlinear
problems. We also improve some related results given in [8,13,7,1] with two objectives. First, the present work is a first step
in the nonlinear backward Cauchy problem, in which the operator A has a continuous spectrum. Thus, for some related
questions on homogeneous parabolic equations backwards in time, as in the case A where has a continuous spectrum, we
refer the reader to [12,13]. As far as we know, there are few nonlinear backward Cauchy problems until now. Second, we give
some new error estimates, which are not of the Holder type. The major object of this paper is to provide a quite simple and
convenient new regularization method. Meanwhile, some more faster convergence error estimates are given. Especially, the
convergence of the approximate solution at t = 0 is also proved.
This paper is organized as follows. In the next section, for ease of the reading, we summarize some well-known facts in
semigroup of operator. The stability estimates of the regularized solution will be presented in Section 2.
2. The basic results
In this section we present the notation and the functional setting which will be used in this paper and prepare some
material which will be used in our analysis.
We denote by {Eλ , λ ≥ 0} the spectral resolution of the identity associated to A.
∞
We denote by S (t ) = e−tA = 0 e−t λ dEλ ∈ L(H ), t ≥ 0, the C0 -semigroup generated by −A. Some basic definitions are
listed in the following theorem.
See Chap. 2, Theorem 6.13, in [14]. For this family of operators we have:
1.

2.
3.
4.
5.

S (t ) ≤ 1, for all t ≥ 0;
the function t −→ S (t ), t > 0, is analytic;
for every real r ≥ 0 and t > 0, the operator S (t ) ∈ L(H , D (Ar ));
for every integer k ≥ 0 and t > 0, S (k) (t ) = Ak S (t ) ≤ c (k)t −k ;
for every x ∈ D (Ar ), r ≥ 0 we have S (t )Ar x = Ar S (t )x.

Theorem 1. Let A : D(A) ⊂ H → H be a self-adjoint operator on the Hilbert space X over K . Then there exists exactly one
spectral family {Eλ } such that
+∞

λdEλ u

Au =

(5)

0

for all u ∈ D(A).
In this connection, u ∈ D(A) iff the integral (5) exists, i.e.,
+∞

λ2 d Eλ u

2

< ∞.

0

Definition. Let A : D(A) ⊂ H → H be a self-adjoint operator on the Hilbert space H over K and let f , g : R → K be a
piecewise continuous function. We set
+∞

D(f (A)) =

|f (λ)|2 d Eλ u

u∈H :

2

<∞

0

and define the linear operator f (A) : D(A) ⊂ H → H by the formula
+∞

f (A)u =

f (λ)dEλ u
0

for all u ∈ D(f (A)).

1968

N.H. Tuan et al. / Nonlinear Analysis 73 (2010) 1966–1972

3. The main results
Now we are ready to state and prove the main results of this paper. For 0 ≤ t ≤ s ≤ T , we denote
Rα (λ, t ) = e−λt (αλ + e−λT )−1 .
This also means that
Rα (λ, T + t − s) = e(s−t −T )λ (αλ + e−λT )−1 .
If the problem (1)–(2) admits a solution u then this solution can be represented by
∞

u(t ) =

∞

T

eλ(T −t ) dEλ ϕ −

eλ(s−t ) dEλ f (u(s))ds.

0

t

0

Since t < T , we know from (6) that, the terms e−(t −T )λ and e−(t −s)λ are the sources of instability. So, we replace them by
approximation terms such as Rα (λ, t ), Rα (λ, T + t − s). Thus, it is easy to see that
lim Rα (λ, t ) = e−(t −T )λ

α→0

and
lim Rα (λ, T + t − s) = e−(t −s)λ .

α→0

Hence, the ill-posed problem (1)–(2)can be approximated by
∞

uα (t ) =

∞

T

Rα (λ, t )dEλ ϕ −
0

Rα (λ, T + t − s)dEλ f (uα (s))ds.
t

(6)

0

Noting that if f = 0, (6) is also the problem (2.2) given in page 2, [13].
Our first main theorem is the following,
Theorem 1. Let 0 < α < Te, ϕ ∈ H and let f : H → H be a continuous operator satisfying
f (w) − f (v) ≤ k w − v ,
for a k > 0 independent of w, v ∈ H , t ∈ R. Then problem (6) has a unique solution uα ∈ C ([0, T ]; H ).
Proof of Theorem 1. First, we consider the following function for λ > 0
Rα (λ, 0) = (αλ + e−λT )−1 .
It is easy to prove that for 0 < α < eT then
Rα (λ, 0) ≤ Rα

ln αT
,0
T

= T α −1 ln(Teα −1 )

−1

.

(7)

For 0 ≤ t ≤ s ≤ T , then
Rα (λ, T + t − s) = exp((s − t − T )λ)(αλ + e−λT )

t −s
T

(αλ + e−λT )

≤ exp((s − t − T )λ)(αλ + e−λT )

t −s
T

(e−λT )

≤ (Rα (λ, 0))
≤α

t −s
T

s−t −T
T

s−t −T
T

s−t
T

T −1 ln(Teα −1 )

t −s
T

= M (α, t )M −1 (α, s)

(8)

where
t

M (α, t ) = α T T −1 ln(Teα −1 )

t
T

−1

,

t ∈ [0, T ].

For w ∈ C ([0, T ]; H ), we define the operator F by
∞

F (w)(t ) =

∞

T

Rα (λ, t )dEλ ϕ −
0

Rα (λ, T + t − s)dEλ f (w(s))ds.
0

(9)

t

Now we prove that for all w, v ∈ C ([0, T ]; H ) the following inequality holds
F m (w)(t ) − F m (v)(t ) ≤

(kT α −1 C )m
|||w − v|||,
m!

(10)


N.H. Tuan et al. / Nonlinear Analysis 73 (2010) 1966–1972

1969

where C = max{T , 1} and |||.||| is sup norm in C ([0, T ]; H ).
In fact, for m = 1, using (8), the Lipschitz property of f and noting that
Rα (λ, T + t − s) ≤ α

t −s
T

t −s
T

T −1 ln(Teα −1 )

<

1

α

,

we have
2

2

∞

T

F (w)(t ) − F (v)(t )

Rα (λ, T + t − s)dEλ (f (w(s)) − f (v(s)))ds

=
t

0

∞

T

T

d Eλ (f (w(s)) − f (v(s)))

(Rα (λ, T + t − s))2 ds

≤

2

T −t

≤

2

2

ds

0

t

T −t

T

f (w(s)) − f (v(s)) ds

α

t

(T − t )k
α

≤

ds

∞

T

d Eλ (f (w(s)) − f (v(s)))

α

=

2

t

t

0

2

w(s) − v(s) 2 ds

≤ (kTC )2 α −2 |||w − v||| |2 .

(11)

Suppose that (11) holds for m = j. We prove that (11) holds for m = j + 1. In fact, we have
F

(w)(t ) − F

(v)(t )

j +1

2

2

∞

T
j +1

Rα (λ, T + t − s)dEλ (f (F w)(s) − f (F v)(s))ds
j

=
t

j

0

(T − t )k 2 T j
F (w)(s) − F j (v)(s) 2 ds
α
t
(kT α −1 C )2j+2
|||w − v|||2 .
≤
(j + 1)!
≤

Therefore, by the induction principle, we have (10) for all w, v ∈ C ([0, T ]; H ). We consider F : C ([0, T ]; H ) → C ([0, T ]; H ).
(kT α −1 C )m

Since limm→∞
= 0, there exists a positive integer number m0 such that F m0 is a contraction. It follows that the
m!
equation F m0 (w) = w has a unique solution uα ∈ C ([0, T ]; H ).
We claim that F (uα ) = uα . In fact, one has F (F m0 (uα )) = F (uα ). Hence F m0 (F (uα )) = F (uα ). By the uniqueness of the
fixed point of F m0 , one has F (uα ) = uα , i.e., the equation F (w) = w has a unique solution uα ∈ C ([0, T ]; H ).
Now we have the following theorem in which, we show that the stability magnitude of our method is less than order one
in the previous methods.
Theorem 2. Let α be as in Theorem 1. Then the (unique) solution of problem (1)–(2) depends continuously (in C ([0, T ]; H )) on
ϕ , this means that, if u and v are two solutions of problem (6) corresponding to the final value ϕ and ω respectively then
u(t ) − v(t ) ≤

√

t
T

t −T
2
2
2ek (T −t ) α T T −1 ln(Teα −1 )

−1

ϕ−ω .

Proof of Theorem 2. It is well known that, for all t ∈ [0, T ],
∞

u(t ) − v(t ) =

∞

T

Rα (λ, t )dEλ (ϕ − ω) −
0

Rα (λ, T + t − s)dEλ (f (u(s)) − f (v(s)))ds.
0

t

Using Lemmas 2 and 3 and the Lipchitz property of f we get
∞

u(t ) − v(t )

2

≤ 2α −2 M 2 (α, t )

d Eλ (ϕ − ω)

2

0

∞

T

T

(Rα (λ, T + t − s))2 ds

+2
0

t

d Eλ (f (w(s)) − f (v(s)))

2

ds

t
T

≤ 2α −2 M 2 (α, t ) ϕ − ω + 2k2 (T − t )

M 2 (α, t )M −2 (α, s) u(s) − v(s)
t

2

ds.


1970

N.H. Tuan et al. / Nonlinear Analysis 73 (2010) 1966–1972

Hence
T

M −2 (α, t ) u(t ) − v(t )

M −2 (α, s) u(s) − v(s)

≤ 2α −2 ϕ − ω + 2k2 (T − t )

2

2

ds.

t

Applying Gronwall’s inequality, we have
M −2 (α, t ) u(t ) − v(t )

2 (T −t )2

2

α −2 ϕ − ω 2 .

≤ 2e2k

Hence, we get
u(t ) − v(t ) ≤

√

t −T
2
2
2ek (T −t ) α T T −1 ln(Teα −1 )

t
T

−1

ϕ−ω .

This inequality follows the solution of the problem (6) depending continuously on ϕ and Theorem 2 is proved.
Remark 1. In [7] (see p. 238), and in [1], the authors give better stability estimates than the latter discussed methods. They
t
T

show that the stability estimate is of order
T
having the form of
T .

−1

. In [13] (see Theorem 2.2 page 2), the authors give another stability bound

α(1+ln α )

In our paper, we give a better estimation of the stability order, which is
t

C α T −1 T −1 ln(Teα −1 )

t
T

−1

.

Comparing our results with the above related results, we see that the order of the error, introduced by small changes in the
final value ϕ , is less than the order given in [4,8,13,7,1]. This is among of the best advantages of our method.
Theorem 3. Let u ∈ C ([0, T ]; H ) be a solution of (1)–(2). Assume that u has the eigenfunction expansion u(t ) =
satisfying
∞

λ2 e2t λ d Eλ u(t )

2

< ∞,

∞
0

dEλ u(t )

(12)

0

for every t ∈ (0, T ].
Then, for any α ∈ (0, Te),
k2 T 2

u(t ) − uα (t ) ≤ N exp

t

α T T −1 ln(Teα −1 )

2

t
T

−1

,

∀t ∈ (0, T ],

where
∞

N = sup

λ2 e2t λ d Eλ u(t ) 2 ,

2

t ∈[0,T ]

0

and uα is the unique solution of Problem (6).
Proof of Theorem 3. The function u(t ), uα (t ) has the expansion
∞

u(t ) =
0

∞

T

eλ(T −t ) dEλ ϕ −
t

0
T

∞

uα (t ) =

∞

Rα (λ, t )dEλ ϕ −
0

eλ(s−t ) dEλ f (u(s))ds,
Rα (λ, T + t − s)dEλ f (uα (s))ds.

t

0

Hence, we get
∞

u(t ) − uα (t ) =

(eT λ −

0

αλ +

e−λT

) e−λt dEλ ϕ −
0

Rα (λ, T + t − s)dEλ (f (uα (s)) − f (u(s))) ds

+
t

∞

=

0

∞

T

αλRα (λ, t ) eT λ dEλ ϕ −

0

t

esλ dEλ f (u(s))ds

0

∞

T

Rα (λ, T + t − s)dEλ (f (uα (s)) − f (u(s))) ds

+
t

∞
0

t

eλ(s−t −T ) dEλ f (u(s))ds

∞

T

=

∞

T

1

0

∞

T

α Rα (λ, t )λet λ dEλ u(t ) +

αλRα (λ, T + t − s)dEλ (f (uα (s)) − f (u(s))) ds
t

0

(13)


N.H. Tuan et al. / Nonlinear Analysis 73 (2010) 1966–1972

1971

then
∞

u(t ) − uα (t )

2

≤ 2M 2 (α, t )

λ2 e2t λ d Eλ u(t )

2

0

∞

T

+ (T − t )M 2 (α, t )
∞

2

ds

0

t

≤ 2M 2 (α, t )

d Eλ (f (w(s)) − f (v(s)))

M −2 (α, s)

λ2 e2t λ d Eλ u(t )

T
2

M −2 (α, s) (f (w(s)) − f (v(s)))

+ (T − t )M 2 (α, t )

2

ds.

t

0

So, we obtain
T

M −2 (α, t ) u(t ) − uα (t )

2

M −2 (α, s) u(s) − uα (s) ds.

≤ N + k2 (T − t )
t

Using Gronwall’s inequality, we get
u(t ) − uα (t )

2

2T 2

≤ N 2 ek

2t
T

2t

α T T −1 ln(Teα −1 )

−2

.

This completes the proof of Theorem 3.
Remark 2. 1. One superficial advantage of this method is that there is an error estimation in the original time t = 0, which
is not given in many recently known results in [7,1]. We have the following estimate

u(0) − uα (0) ≤ NT ekT

1 + ln

T

α

−1

.

This error is similar to Theorem 2.6, page 5 in [13].
2. If f (u) = 0, we have
∞

u(0) =

et λ dEλ u(t ).

0

Taking the derivative of u, we get
∞

u (0) = −

λe−t λ dEλ u(t ).

0

Then, we get
∞

u (0)

2

= Au(0)

2

=

λ2 e2λt d Eλ u(t ) 2 .

0

Hence, the condition (12) is natural and acceptable.
3. In [7] (see p. 241), and in [1] the error estimates between the exact solution and the approximate solution are of
t

the same order U (α, t ) = C α T . And in this article, the convergence rate is in a slightly different form than given in [4,
t

t

−1

V (α,t )

8,13,1], defined by V (α, t ) = Dα T T −1 ln(Teα −1 ) T . We note that limα→0 U (α,t ) = 0. Hence, this error is the optimal
error estimate to our knowledge. Moreover, we also have limα→0 (limt →0 U (α, t )) = C and limα→0 (limt →0 V (α, t )) =
limα→0 D T −1 ln(Teα −1 )

−1

= 0. So, it is easy to see that if the time t is near to the original time t = 0, the convergence

of the approximation solution is very slow. This implies that the methods such as Quasi-boundary value and stabilized quasireversibility studied in [7,1], are not useful to derive the error estimations in the case where t is in the neighborhood of zero.
This also proves that our method give a better approximation than the previous case which we know. Comparing (13) with
the results obtained in [8,13,7,1], we realize that (13) is sharp and the best known estimate. This is a generalization of many
previous results.
Acknowledgements
The authors would like to thank Professor Ravi P. Agarwal for his valuable help in the presentation of this paper. The
authors are also grateful to the anonymous referees for their valuable comments leading to the improvement of our paper.
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